On densities and gaps (Q1090708)

The authors investigate certain relations between the asymptotic densities of subsets of a set \(A=\{m_ 1,m_ 2,...\}\) of positive integers and the lengths of gaps in A by means of the ''analytical'' properties of the two-dimensional set \(S(A)=\{(\underline d(B),\bar d(B))\in {\mathbb{R}}^ 2\); \(B\subseteq A\}\), where ḏ(B) and \(\bar d(B)\) denote the lower and upper asymptotic density of B. A special role here is played by the function \(f_ S\), whose graph is the lower part of the boundary of S(A). So for instance, the right derivatives \(D^+_ f(0)\) of f at the origin is not less than \(\lambda (A)=\limsup_{j\to \infty}m_{j+1}/m_ j\). Conversely, given a set \(S\subseteq {\mathbb{R}}^ 2\) satisfying certain conditions and a real number \(\lambda\) such that \(1\leq \lambda \leq D^+_ f(0)\), where \(f=f_ S\) then there exists a set A of positive integers such that \(S(A)=S\) and \(\lambda (A)=\lambda\). They also give a characterization of \(D^+_ f(0)\) of the function f corresponding to a given set A in terms of the lengths of intervals in which the set A possesses ''few'' elements, etc.